PHYSICS AND FRACTAL STRUCTURES docx

2 1.3 Metric properties: Hausdorff dimension, topological dimension .... 1.2 The notion of dimension A common method of measuring a length, a surface area or a volumeconsists in covering

AND FRACTAL STRUCTURES

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J.-F GOUYET

Laboratoire de Physique de la Matière Condensée

Ecole Polytechnique

When intellectual and political movements ponder their roots, no event loomslarger than the first congress The first meeting on fractals was held in July

1982 in Courchevel, in the French Alps, through the initiative of Herbert Buddand with the support of IBM Europe Institute Jean-François Gouyet’s bookreminds me of Courchevel, because it was there that I made the acquaintanceand sealed the friendship of one of the participants, Bernard Sapoval, and it wasfrom there that the fractal bug was taken to Ecole Polytechnique Sapoval,Gouyet and Michel Rosso soon undertook the work that made their laboratory

an internationally recognized center for fractal research If I am recounting allthis, it is to underline that Gouyet is not merely the author of a new textbook,but an active player on a world-famous stage While the tone is straightforward,

as befits a textbook, he speaks with authority and deserves to be heard

The topic of fractal diffusion fronts which brought great renown to Gouyetand his colleagues at Polytechnique is hard to classify, so numerous and variedare the fields to which it applies I find this feature to be particularly attractive.The discovery of fractal diffusion fronts can indeed be said to concern thetheory of welding, where it found its original motivation But it can also be said

to concern the physics of (poorly) condensed matter Finally it also concernsone of the most fundamental concepts of mathematics, namely, diffusion Eversince the time of Fourier and then of Bachelier (1900) and Wiener (1922), thestudy of diffusion keeps moving forward, yet entirely new questions comeabout rarely Diffusion fronts brought in something entirely new

Returning to the book itself, if the variety of the topics comes as a surprise

to the reader, and if the brevity of some of treatments leaves him or her hungryfor more, then the author will have achieved the goal he set himself The most

important specialized texts treating the subject are carefully referenced andshould satisfy most needs.

To sum up, I congratulate Jean-François warmly and wish his book thegreat success it deserves

Benoît B MANDELBROT

Yale University

IBM T.J Watson Research Center

Hath smaller Fleas that on him prey, And these have smaller yet to bite ‘em And so proceed ad infinitum.

…

Jonathan Swift, 1733,

On poetry, a Rhapsody

Foreword v

Preface xi

1 Fractal geometries 1.1 Introduction 1

1.2 The notion of dimension 2

1.3 Metric properties: Hausdorff dimension, topological dimension 4

1.3.1 The topological dimension 5

1.3.2 The Hausdorff–Besicovitch dimension 5

1.3.3 The Bouligand–Minkowski dimension 6

1.3.4 The packing dimension 8

1.4 Examples of fractals 10

1.4.1 Deterministic fractals 10

1.4.2 Random fractals 19

1.4.3 Scale invariance 21

1.4.4 Ambiguities in practical measurements 22

1.5 Connectivity properties 23

1.5.1 Spreading dimension, dimension of connectivity 23

1.5.2 The ramification R 25

1.5.3 The lacunarity L 25

1.6 Multifractal measures 26

1.6.1 Binomial fractal measure 27

1.6.2 Multinomial fractal measure 30

1.6.3 Two scale Cantor sets 36

1.6.4 Multifractal measure on a set of points 38

2 Natural fractal structures: From the macroscopic… 2.1 Distribution of galaxies 41

2.1.1 Distribution of clusters in the universe 42

2.1.2 Olbers’ blazing sky paradox 43

2.2 Mountain reliefs, clouds, fractures… 45

2.2.1 Brownian motion, its fractal dimension 46

2.2.2 Scalar Brownian motion 48

2.2.3 Brownian function of a point 49

2.2.4 Fractional Brownian motion 49

2.2.5 Self-affine fractals 52

2.2.6 Mountainous reliefs 57

2.2.7 Spectral density of a fractional Brownian motion, the spectral exponent ß 58

2.2.8 Clouds 61

2.2.9 Fractures 62

2.3 Turbulence and chaos 65

2.3.1 Fractal models of developed turbulence 66

2.3.2 Deterministic chaos in dissipative systems 72

3 Natural fractal structures: …to the microscopic 3.1 Disordered media 89

3.1.1 A model: percolation 89

3.1.2 Evaporated films 105

3.2 Porous media 107

3.2.1 Monophasic flow in poorly connected media 108

3.2.2 Displacement of a fluid by another in a porous medium 109

3.2.3 Quasistatic drainage 111

3.3 Diffusion fronts and invasion fronts 118

3.3.1 Diffusion fronts of noninteracting particles 118

3.3.2 The attractive interaction case 125

3.4 Aggregates 130

3.4.1 Definition of aggregation 130

3.4.2 Aerosols and colloids 132

3.4.3 Macroscopic aggregation 140

3.4.4 Layers deposited by sputtering 141

3.4.5 Aggregation in a weak field 142

3.5 Polymers and membranes 146

3.5.1 Fractal properties of polymers 146

3.5.2 Fractal properties of membranes 152

4 Growth models 4.1 The Eden model 157

4.1.1 Growth of the Eden cluster: scaling laws 159

4.1.2 The Williams and Bjerknes model 163

4.1.3 Growing percolation clusters 164

4.2 The Witten and Sander model 165

4.2.1 Description of the DLA model 165

4.2.2 Extensions of the Witten and Sander model 167

4.2.3 The harmonic measure and multifractality 172

4.3 Modeling rough surfaces 174

4.3.1 Self-affine description of rough surfaces 174

4.3.2 Deposition models 174

4.3.3 Analytical approach to the growth of rough surfaces 176

4.4 Cluster–cluster aggregation 177

4.4.1 Diffusion–limited cluster–cluster aggregation 177

4.4.2 Reaction–limited cluster–cluster aggregation 179

4.4.3 Ballistic cluster–cluster aggregation and other models 180

5 Dynamical aspects 5.1 Phonons and fractons 183

5.1.1 Spectral dimension 183

5.1.2 Diffusion and random walks 188

5.1.3 Distinct sites visited by diffusion 191

5.1.4 Phonons and fractons in real systems 192

5.2 Transport and dielectric properties 194

5.2.1 Conduction through a fractal 194

5.2.2 Conduction in disordered media 197

5.2.3 Dielectric behavior of composite media 207

5.2.4 Response of viscoelastic systems 208

5.3 Exchanges at interfaces 211

5.3.1 The diffusion-limited regime 213

5.3.2 Response to a blocking electrode 213

5.4 Reaction kinetics in fractal media 215

6 Bibliography 219

7 Index 231

The introduction of the concept of fractals by Benoît B Mandelbrot at thebeginning of the 1970’s represented a major revolution in various areas ofphysics The problems posed by phenomena involving fractal structures may

be very difficult, but the formulation and geometric understanding of theseobjects has been simplified considerably This no doubt explains the immensesuccess of this concept in dealing with all phenomena in which a semblance ofdisorder appears

Fractal structures were discovered by mathematicians over a century ago

and have been used as subtle examples of continuous but nonrectifiable curves, that is, those whose length cannot be measured, or of continuous but nowhere

differentiable curves, that is, those for which it is impossible to draw a tangent

at any their points Benoît Mandelbrot was the first to realize that many shapes

in nature exhibit a fractal structure, from clouds, trees, mountains, certain plants,rivers and coastlines to the distribution of the craters on the moon Theexistence of such structures in nature stems from the presence of disorder, orresults from a functional optimization Indeed, this is how trees and lungsmaximise their surface/volume ratios

This volume, which derives from a course given for the last three years atthe Ecole Supérieure d’Electricité, should be seen as an introduction to thenumerous phenomena giving rise to fractal structures It is intended forstudents and for all those wishing to initiate themselves into this fascinatingfield where apparently disordered forms become geometry It should also beuseful to researchers, physicists, and chemists, who are not yet experts in thisfield

This book does not claim to be an exhaustive study of all the latestresearch in the field, yet it does contains all the material necessary to allow thereader to tackle it Deeper studies may be found not only in Mandelbrot’sbooks (Springer Verlag will publish a selection of books which bring togetherreprints of published articles along with many unpublished papers), but also inthe very abundant, specialized existing literature, the principal references ofwhich are located at the end of this book

The initial chapter introduces the principal mathematical concepts needed

to characterize fractal structures The next two chapters are given over to fractalgeometries found in nature; the division of these two chapters is intended to

help the presentation Chapter 2 concerns those structures which may extend toenormous sizes (galaxies, mountainous reliefs, etc.), while Chap 3 explainsthose fractal structures studied by materials physicists This classification isobviously too rigid; for example, fractures generate similar structures ranging insize from several microns to several hundreds of meters.

In these two chapters devoted to fractal geometries produced by thephysical world, we have introduced some very general models Thus fractionalBrownian motion is introduced to deal with reliefs, and percolation to deal withdisordered media This approach, which may seem slightly unorthodox seeingthat these concepts have a much wider range of application than the examples towhich they are attached, is intended to lighten the mathematical part of thesubject by integrating it into a physical context

Chapter 4 concerns growth models These display too great a diversityand richness to be dispersed in the course of the treatment of the variousphenomena described

Finally, Chap 5 introduces the dynamic aspects of transport in fractalmedia Thus it completes the geometric aspects of dynamic phenomenadescribed in the previous chapters

I would like to thank my colleagues Pierre Collet, Eric Courtens, FrançoisDevreux, Marie Farge, Max Kolb, Roland Lenormand, Jean-Marc Luck,Laurent Malier, Jacques Peyrière, Bernard Sapoval, and Richard Schaeffer, forthe many discussions which we have had during the writing of this book Ithank Benoît Mandelbrot for the many improvements he has suggestedthroughout this book and for agreeing to write the preface I am especiallygrateful to Etienne Guyon, Jean-Pierre Hulin, Pierre Moussa, and MichelRosso for all the remarks and suggestions that they have made to me and forthe time they have spent in checking my manuscript Finally, I would like tothank Marc Donnart and Suzanne Gouyet for their invaluable assistance duringthe preparation of the final version

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The success of the French original version published by Masson, hasmotivated Masson and Springer to publish the present English translation I amgreatly indebted to them I acknowledge Dr David Corfield who carried outthis translation and Dr Clarissa Javanaud and Prof Eugene Stanley for manyvaluable remarks upon the final translation During the last four years, the use

of fractals has widely spread in various fields of science and technology, andsome new approaches (such as wavelets transform) or concepts (such as scalerelativity) have appeared But the essential of fractal knowledge was alreadypresent at the end of the 1980s

Palaiseau, July 1995

Fractal Geometries

1.1 Introduction

The end of the 1970s saw the idea of fractal geometry spread into numerous

areas of physics Indeed, the concept of fractal geometry, introduced by B.Mandelbrot, provides a solid framework for the analysis of natural phenomena

in various scientific domains As Roger Pynn wrote in Nature, “If this opinion

continues to spread, we won’t have to wait long before the study of fractalsbecomes an obligatory part of the university curriculum.”

The fractal concept brings many earlier mathematical studies within a

single framework The objects concerned were invented at the end of the 19th

century by such mathematicians as Cantor, Peano, etc The term “fractal” was

introduced by B Mandelbrot (fractal, i.e., that which has been infinitely divided,

from the Latin “fractus,” derived from the verb “frangere,” to break) It is

difficult to give a precise yet general definition of a fractal object; we shalldefine it, following Mandelbrot, as a set which shows irregularities on allscales

Fundamentally it is its geometric character which gives it such great scope;

fractal geometry forms the missing complement to Euclidean geometry andcrystalline symmetry.1 As Mandelbrot has remarked, clouds are not spheres,nor mountains cones, nor islands circles and their description requires adifferent geometrization

As we shall show, the idea of fractal geometry is closely linked to

properties invariant under change of scale: a fractal structure is the same “from

near or from far ” The concepts of self-similarity and scale invariance

appeared independently in several fields; among these, in particular, are criticalphenomena and second order phase transitions.2 We also find fractalgeometries in particle trajectories, hydrodynamic lines of flux, waves,landscapes, mountains, islands and rivers, rocks, metals, and compositematerials, plants, polymers, and gels, etc

1 We must, however, add here the recent discoveries about quasicrystalline symmetries.

2 We shall not refer here to the wide and fundamental literature on critical phenomena, renormalization, etc.

Many works on the subject have been published in the last 10 years Basicworks are less numerous: besides his articles, B Mandelbrot has publishedgeneral books about his work (Mandelbrot, 1975, 1977, and 1982); the books

by Barnsley (1988) and Falconer (1990) both approach the mathematicalaspects of the subject Among the books treating fractals within the domain ofthe physical sciences are those by Feder (1988) and Vicsek (1989) (whichparticularly concentrates on growth phenomena), Takayasu (1990), or LeMéhauté (1990), as well as a certain number of more specialized (Avnir, 1989;Bunde and Havlin, 1991) or introductory monographs on fractals (Sapoval,1990) More specialized reviews will be mentioned in the appropriate chapters

1.2 The notion of dimension

A common method of measuring a length, a surface area or a volumeconsists in covering them with boxes whose length, surface area or volume istaken as the unit of measurement (Fig 1.2.1) This is the principle which liesbehind the use of multiple integration in calculating these quantities

d=0 d=1 d=2 d=3

Fig 1.2.1 Paving with lines, surfaces, or volumes.

If 2 is the side (standard length) of a box and d its Euclidean dimension, themeasurement obtained is

M = N 2d = Nµ,where µ is the unit of measurement (length, surface area, or volume in thepresent case, mass in other cases) Cantor, Carathéodory, Peano, etc showedthat there exist pathological objects for which this method fails Themeasurement above must then be replaced, for example, by the 1-dimensionalHausdorff measure This is what we shall now explain

The length of the Brittany’s coastline

Imagine that we would like to apply the preceding method to measure thelength, between two fixed points, of a very jagged coastline such as that of

Brittany.3 We soon notice that we are faced with a difficulty: the length L

depends on the chosen unit of measurement 2 and increases indefinitely as 2decreases (Fig 1.2.2)!

Fig 1.2.2 Measuring the length of a coastline in relation to different units.

For a standard unit 21 we get a length N1 21, but a smaller standardmeasure, 22, gives a new value which is larger,

L (21) = N1 21

L (22) = N2 22 1 L (21)

…and this occurs on scales going from several tens of kilometers down to a fewmeters L.F Richardson, in 1961, studied the variations in the approximatelength of various coastlines and noticed that, very generally speaking, over a 1

Log (Length of the unit measure in kilometres)

Coast of Australia

Land border of Portugal

West coast of Engl

large range of L (2), the length follows a power law4 in 2,

L (2) = N(2) 2 4 2– 3

Figure 1.2.3 shows the behavior of various coastlines as functions of theunit of measurement We can see that for a “normal” curve like the circle, thelength remains constant (3 = 0) when the unit of measurement becomes smallenough in relation to the radius of curvature The dimension of the circle is ofcourse D = 1 (and corresponds to 3 = 0) The other curves display a positiveexponent 3 so that their length grows indefinitely as the standard lengthdecreases: it is impossible to give them a precise length, they are said to benonrectifiable.5 Moreover, these curves also prove to be nondifferentiable.The exponent (1+3) of 1/N(2) defined above is in fact the “fractal

dimension” as we shall see below This method of determining the fractal size

by covering the coast line with discs of radius 2 is precisely the one used byPontrjagin and Schnirelman (1932) (Mandelbrot, 1982, p 439) to define the

covering dimension. The idea of defining the dimension on the basis of acovering ribbon of width 22 had already been developed by Minkowski in

1901 We shall therefore now examine these methods in greater detail

Generally speaking, studies carried out on fractal structures rely both onthose concerning nondifferentiable functions (Cantor, Poincaré, and Julia) and

on those relating to the measure (dimension) of a closed set (Bouligand,Hausdorff, and Besicovitch)

1.3 Metric properties: Hausdorff dimension,

topological dimension

Several definitions of fractal dimension have been proposed Thesemathematical definitions are sometimes rather formal and initially not alwaysvery meaningful to the physicist For a given fractal structure they usually givethe same value for the fractal dimension, but this is not always the case Withsome of these definitions, however, the calculations may prove easier or moreprecise than with others, or better suited to characterize a physical property.Before giving details of the various categories of fractal structures, we shallgive some mathematical definitions and various methods for calculatingdimensions; for more details refer to Tricot’s work (Tricot, 1988), or toFalconer’s books (Falconer, 1985, 1990)

First, we remark that to define the dimension of a structure, this structuremust have a notion of distance (denoted Ix-yI) defined on it between any two ofits points This hardly poses a problem for the structures provided by nature

We should also mention that in these definitions there is always a passage

to the limit 4¯0 For the actual calculation of a fractal dimension we are led todiscretize (i.e., to use finite basic lengths 4): the accuracy of the calculation thendepends on the relative lengths of the unit 4, and that of the system (Sec 1.4.4)

1.3.1 The topological dimension d T

If we are dealing with a geometric object composed of a set of points, wesay that its fractal dimension is dT = 0; if it is composed of line elements, dT8=

1, surface elements dT = 2, etc

“Composed” means here that the object is locally homeomorphic to a point, a line, a surface The topological dimension is invariant under invertible, continuous, but not necessarily differentiable, transformations (homeomorphisms) The dimensions which we shall be speaking of are invariant under differentiable transformations (dilations).

A fractal structure possesses a fractal dimension strictly greater than itstopological dimension

1.3.2 The Hausdorff–Besicovitch dimension,

or covering dimension: dim(E)

The first approach to finding the dimension of an object, E, follows theusual method of covering the object with boxes (belonging to the space inwhich the object is embedded) whose measurement unit µ = 4d(E), where d(E) isthe Euclidean dimension of the object When d(E) is initially unknown, onepossible solution takes µ8= 43 as the unit of measurement for an unknownexponent 3 Let us consider, for example, a square (d = 2) of side L, and cover

it with boxes of side 4 The measure is given by M = Nµ, where N is the

number of boxes, hence N = (L/4)d Thus,

M = N 43 = (L/4)d 43 = L2 4312

If we try 3 = 1, we find that M 7 6 when 4 7 0: the “length” of a square

is infinite If we try 3 = 3, we find that M 7 0 when 4 7 0: the “volume” of a

square is zero The surface area of a square is obtained only when 3 = 2, andits dimension is the same as that of a surface d = 3 = 2

The fact that this method can be applied for any real 3 is very interesting

as it makes possible its generalization to noninteger dimensions

We can formalize this measure a little more First, as the object has nospecific shape, it is not possible, in general, to cover it with identical boxes ofside 4 But the object E may be covered with balls Vi whose diameter (diam Vi)

is less than or equal to 4 This offers more flexibility, but requires that theinferior limit of the sum of the elementary measures be taken as µ =(diam8Vi)3

Therefore, we consider what is called the 3–covering measure (Hausdorff,1919; Besicovitch, 1935) defined as follows:

m3(E) = lim 470 inf{ 5(diam Vi)3 : &Vi 'E, diam Vi 3 4}, (1.3-1)

and we define the Hausdorff (or Hausdorff–Besicovitch) dimension: dim E by

dim E = inf { 3 : m3(E) = 0 }

= sup { 3 : m3(E) = 2} (1.3-2)The Hausdorff dimension is the value of 3 for which the measure jumps fromzero to infinity For the value 3 = dim E, this measure may be anywherebetween zero and infinity

The function m3(E) is monotone in the sense that if a set F is included in E,

E ' F, then m3(E) 4 m3(F) whatever the value of 3.

1.3.3 The Bouligand–Minkowski dimension

We can also define a dimension known as the Bouligand–Minkowskidimension (Bouligand, 1929; Minkowski, 1901), denoted 7(E) Here are somemethods of calculating 7(E):

The Minkowski sausage (Fig 1.3.1)

Let E be a fractal set embedded in a d-dimensional Euclidean space (moreprecisely E is a closed subset of Rd) Now let E(4) be the set of points in Rd at

a distance less than 4 from E E(4) now defines a Minkowski sausage: it is alsocalled a thickening or dilation of E as in image analysis It may be defined asthe union

E(4) = & B4(x), x(E

where B4(x) is a ball of the d-dimensional Euclidean space, centered at x and ofradius 4 We calculate,

1(E) = lim230 (d –log Vold[E(2)]

where Vold simply represents the volume in d dimensions (e.g., the usuallength, surface area, or volume) If the limit exists, 7(E) is, by definition, the

Bouligand–Minkowski dimension.

Naturally, we recover from this the usual notion of dimension: let us take

as an example a line segment of length L The associated Minkowski sausagehas as volume Vold (E),

in d = 2 : 24 L + 642,

in d = 3 : 642 L + (46/3)43, 1

Fig 1.3.1 Minkowski sausage or thickening of a curve E.

so that neglecting higher orders in 4, Vold (E) 8 4 d – 1

In general terms we have:

If E is a point: Vold (E) 8 4d, 2(E) = 0

If E is a rectifiable arc: Vold (E) 8 4d – 1, 2(E) = 1

If E is a k-dimensional ball: Vold (E) 8 4d – k, 2(E) = k

In practice, 7(E) is obtained as the slope of the line of least squares of theset of points given by the plane coordinates,

{ log 1/4, log Vold [E(4) /4d ] }

This method is easy to use The edge effects (like those obtained above inmeasuring a segment of length L) lead to a certain inaccuracy in practice (i.e., to

a curve for values of 4 which are not very small)

The box-counting method (Fig 1.3.2)

This is a very useful method for many fractal structures Let N(4) be thenumber of boxes of side 4 covering E:

1(E) = lim230 (log N(2)– log 2 ) (1.3-4) 1

Fig.1.3.2 Measurement of the dimension of a curve by the box-counting method.

The box-counting method is commonly used, particularly for self-affinestructures (see Sec 2.2.5)

The dimension of a union of sets is equal to the largest of the dimensions of these sets: 7(E&F) = max {7(E), 7(F)}.

The limit 7(E) may depend on the choice of paving If there are two different limits Sup and Inf, the Sup limit should be taken.

The disjointed balls method (Fig 1.3.3)

Let N(4) be the maximum number of disjoint balls of radius 4 centered on

the set E: then

2(E) = lim 470 log N(4) / |log 4| (1.3-5)This method is rarely used in practice

1

Fig.1.3.3 Measuring the dimension of a curve by the disjointed balls method.

The dividers’ method (Richardson, 1960)

This is the method we described earlier (Fig 1.2.2).1

1

1

1

2

Let N(4) be the number of steps of length 4 needed to travel along E:

2(E) = lim 470 log N(4) / |log 4| (1.3-6)

Notice that all the methods give the same fractal dimension, 7(E), when it exists (see Falconer, 1990), because we are in a finite dimensional Euclidean space This is

no longer true in an infinite dimensional space, (function space, etc.).

1.3.4 The packing dimension [or Tricot dimension: Dim (E)]

Unlike the Hausdorff–Besicovitch dimension, which is found using the dimensional Hausdorff measure, the box-counting dimension 7(E) is notdefined in terms of measure This may lead to difficulties in certain theoreticaldevelopments This problem may be overcome by defining the packingdimension, following similar ideas to those of the 3-dimensional Hausdorffmeasure (Falconer, 1990) Let {Vi} be a collection of disjoint balls, and

3-Po3(E) = lim 470 sup{ 5(diam Vi)3, diam Vi 3 4}

As this expression is not always a measure we must consider

P1(E) = inf{ Po1(Ei) : 2 1

i = 1 Ei3 E}

.The packing dimension is defined by the following limit:

Dim E = sup{3 : P 3(E) = 2 } = inf {3 : P 3(E) = 0 }, (1.3-7a)alternatively, according to the previous definitions:

Dim E = inf{sup 2(Ei) : & Ei ' E} (1.3-7b)

The following inequalities between the various dimensions defined above are always true:

dim E 3 Dim E 3 2 (E) dim E + dim F 3 dim E9F

3 dim E + Dim F

3 Dim E9F 3 Dim E + Dim F.

Notice that for multifractals box-counting dimensions are in practice rather Tricot dimensions.

Other methods of calculation have been proposed by Tricot (Tricot, 1982)which could prove attractive in certain situations Without entering into the

details, we should also mention the method of structural elements, the method

of variations and the method of intersections.

Theorem: If there exists a real D and a finite positive measure µ such that for

all x(E, (Br(x) being the ball of radius r centered at x),

log µ[Br(x)]/log r 7 D, then

physics when the above theorem applies.

So we now have the following relation giving the mass inside a ball ofradius r,

M = µ(Br (x)) 1 r D , (1.3-10)where the center x of the ball B is inside the fractal structure E.

We shall of course take the physicist’s point of view and not burden

ourselves, at first, with too much mathematical rigor The fractal dimension will

in general be denoted D and, in the cases considered, we shall suppose that,unless specified otherwise, the existence theorem applies and therefore that thefractal dimension is the same for all the methods described above

Units of measure

The above relation can often be written in the form of a dimensionlessequation, by introducing the unit of length 4u and volume (4u)d or mass 5u =(4u)d 5 (by assuming a uniform density 5 over the support):

Examples of this for the Koch curve and the Sierpinski gasket will begiven later in Sec 1.4.1 In this case the unit of volume is that of a space withdimension equal to the topological dimension of the geometric objects making

up the set (see Sec 1.3.1)

From a strictly mathematical point of view the term “dimension” should be reserved for sets For measures, we can think of the set covered by a uniform measure However, we can define the dimension of a measure by

dim(µ) = inf { dim(A), µ(Ac)=0 },

A being a measurable set and Ac its complement This dimension is often strictly less than the dimension of the support This happens with the information dimension described in Sec 1.6.2.

For objects with a different scaling factor in different spatial directions, the counting dimension differs from the Hausdorff dimension (see Fig 2.2.8).

box-Having defined the necessary tools for studying fractal structures, it is nowtime to get to the heart of the matter by giving the first concrete examples offractals

1.4 Examples of fractals

1.4.1 Deterministic fractals

Some fractal structures are constructed simply by using an iterativeprocess consisting of an initiator (initial state) and a generator (iterativeoperation)

The triadic Von Koch curve (1904)

Each segment of length ε is replaced by a broken line (generator),

composed of four segments of length ε/3, according to the following recurrencerelation: 

(generator)

At iteration zero, we have an initiator which is a segment in the case of the

triadic Koch curve, or an equilateral triangle in the case of the Koch island Ifthe initiator is a segment of horizontal length L, at the first iteration (the curvecoincides with the generator) the base segments will have length ε1 = L/3;

at the second iteration they will have length ε2 = L/9 as each segment is againreplaced by the generator, then ε3 = L/33 at the third iteration

,and so on The relations giving the length L of the curve are thus

Ln= LD (εn)1–D where D = log 4 / log 3 = 1.2618…

For a fixed unit length εn, Ln grows as the Dth power of the size L of the curve.Notice that here again we meet the exponent ρ = D–1 of εn, which we first met

in Sec 1.2 (Richardson's law) and which shows the divergence of Ln as εn →0

At a given iteration, the curve obtained is not strictly a fractal but according to Mandelbrot’s term a “prefractal” A fractal is a mathematical object obtained in the limit of a series of prefractals as the number of iterations n tends to infinity In everyday language, prefractals are often both loosely called “fractals”.

The previous expression is the first example given of a scaling law whichmay be written

Ln / εn = f (L /εn ) = ( L /εn )D . (1.4-1)

A scaling law is a relation between different dimensionless quantities

describing the system, (the relation here is a simple power law) Such a law isgenerally possible only when there is a single independent unit of length in theobject (here εn)

A structure associated with the Koch curve is obtained by choosing anequilateral triangle as initiator The structure generated in this way is the well-

known Koch island (see Fig 1.4.1).

Fig 1.4.1 Koch island after only three iterations Its coastline is fractal, but the         island itself has dimension 2 (it is said to be a surface fractal).

Simply by varying the generator, the Koch curve may be generalized togive curves with fractal dimension 1≤D ≤ 2 A straightforward example isprovided by the modified Koch curve whose generator is

α

and whose fractal dimension is D = log 4 / log [2 + 2sin(α/2)] Notice that inthe limit α = 0 we have D = 2, that is to say a curve which fills a triangle It isnot exactly a curve as it has an infinite number of multiple points But theconstruction can be slightly modified to eliminate them The dimension D = 2

(= log 9/log 3) is also obtained for the Peano curve (Fig 1.4.2) (which is

dense in a square) whose generator is formed from 9 segments with a change

by a factor 3 in the linear dimension, i.e.,

.This gives after the first three iterations,

Fig 1.4.2 First three iterations of the Peano curve (for graphical reasons the scale is simultaneously dilated at each iteration by a factor of 3) The Peano curve is dense in

the plane and its fractal dimension is 2.

This construction has also been modified (rounding the angles) to eliminatedouble points

The von Koch and Peano curves are as their name indicates: curves, that is,

their topological dimension is

dT = 1

Practical determination of the fractal dimension

using the mass-radius relation

As mentioned earlier, a method which we shall be using frequently todetermine fractal dimensions6 consists in calculating the mass of the structurewithin a ball of dimension d centered on the fractal If the embedding space isd-dimensional, and of radius R, then

M ∝ RD .The measure here is generally a mass, but it could equally well be a “surfacearea” or any other scalar quantity attached to the support (Fig 1.4.3)

6 The box-counting method will also be frequently used.

In the case of the Koch curve, we could check to see that D = log 4/log 3,

as is the case for the different methods shown above Notice that if the εn arenot chosen in the sequence εn = L/3n, the calculations prove much morecomplicated, but the limit as ε → 0 still exists and gives D

R

Fig 1.4.3 Measuring the fractal dimension of a Koch curve using the relationship bet-ween mass and radius If each segment represents a unit of “fractal surface area” (1 cm D , say), the “surface area” above is equal to 2 cm D when R = 1 cm, 8 cm D when

Direct determination of the fractal dimension and

the multiscale case

The fractal dimension D may be found directly from a single iteration if

the limit structure is known to be a fractal If a fractal structure of size L with

mass M (L) = A(L) LD gives after iteration k elements of size L/h, we then have

an implicit relation in D:

M(L) = k M (L/h), hence A(L) LD = kA(L/h) (L/h)D

D is thus determined asymptotically (L→∞) by noticing that

A(L/h) / A(L) → 1 as L → ∞ Hence k (1/h)D = 1

For example, the Koch curve corresponds to k = 4 and h = 3 Moreover,A(L) is independent of L here

Later on we shall meet multiscale fractals, giving at each iteration ki

elements of size L/hi (i = 1,…, n) Thus

M (L) =k1M (L/h1) + k2M (L/h2) + + knM (L/hn),

which means that the mass of the object of linear size L is the sum of ki masses

of similar objects of size L/hi Thus,

k1(1/h1)D + k2(1/h2)D… + kn(1/hn)D = 1, (1.4-2)which determines D

Cantor sets

These are another example of objects which had been much studied beforethe idea of fractals was introduced The following Cantor set is obtained byiteratively deleting the central third of each segment:

⇒

⇒

initiator generator

Fig 1.4.4 Construction of the first five iterations of a Cantor set In order to have a clearer representation and to introduce the link between measure and set, the segments have been chosen as bars of fixed width (Cantor bars), consequently representing a uniform density distributed over the support set (uniform measure, see also Sec.1.6.3) In this way the fractal dimension and the mass dimension are identified.

Five iterations are shown in Fig 1.4.4

The fractal dimension of this set is

D = log 2/ log 3 = 0.6309

For Cantor sets we have 0 < D < 1: it is said to be a “dust.” As it is composed

only of points, its topological dimension is dT = 0

To demonstrate the fact that the fractal dimension by itself does notuniquely characterize the object, we now construct a second Cantor set with thesame fractal dimension but a different spatial structure (Fig 1.4.5): at eachiteration, each element is divided into four segments of length 1/9, which isequivalent to uniformly spacing the elements of the second iteration of theprevious set In fact these two sets differ by their lacunarity (cf Sec 1.5.3), that

is, by the distribution of their empty regions

Fig 1.4.5 Construction of the first two iterations of a different Cantor set having

the same fractal dimension.

Mandelbrot–Given curve

Iterative deterministic processes have shown themselves to be of greatvalue in the study of the more complex fractal structures met with in nature,since their iterative character often enables an exact calculation to be made TheMandelbrot-Given curve (Mandelbrot and Given, 1984) is an instructiveexample of this as it simulates the current conducting cluster of a network ofresistors close to their conductivity threshold (a network of resistors so many

of which are cut that the network barely conducts) It is equally useful forunderstanding multifractal structures (see Fig 1.4.6) We shall take this upagain in Sec 5.2.2 (hierarchical models) as it is a reasonable model for the

“backbone” of the infinite percolation cluster (Fig 3.1.8)

The generator and first two iterations are as follows:

Fig 1.4.6 Construction of the first three iterations of a Mandelbrot–Given set This fractal has a structure reminiscent of the percolation cluster which plays an important

role in the description of disordered media (Sec 3.1).

The vertical segments of the generator are slightly shortened to avoiddouble points The fractal dimension (neglecting the contraction of the verticalsegments) is D = log 8/ log 3 ≅ 1.89…

“Gaskets” and “Carpets”

These structures are frequently used to carry out exact, analyticcalculations of various physical properties (conductance, vibrations, etc.)

Sierpinski gasket

Fig 1.4.7 Iteration of the Sierpinski gasket composed of full triangles (the object is

made up only of those parts left coloured black).

The scaling factor of the iteration is 2, while the mass ratio is 3 (see Fig 1.4.7).The corresponding fractal dimension is thus

D = log 3/log 2 = 1.585…

The Sierpinski gasket generated by the edges only is also often used (seeFig 1.4.8) It clearly has the same fractal dimension D = log 3/log 2 =1.585…

Fig 1.4.8 Iteration of the Sierpinski gasket composed of sides of triangles.

The two structures can be shown to “converge” asymptotically towardsone other, in the sense of the Hausdorff distance (see, e.g., Barnsley, 1988)

Sierpinski carpet

Fig 1.4.9 Iteration of a Sierpinski carpet.

The scaling factor is 3 and the mass ratio (black squares) is 8 (see Fig.1.4.9) Hence,

D = log 8/log 3 = 1.8928…

Other examples

Examples of deterministic fractal structures constructed on the basis of theSierpinski gasket and carpet can be produced endlessly These geometries can

prove very important for modelling certain transport problems in porousobjects or fractal electrodes Here are a couple of three-dimensional examples,the 3d gasket and the so-called Menger sponge (Fig 1.4.10).

3d gasket (Mandelbrot) Menger sponge

D = log 4/log 2 = 2 D = log 20/log 3 ≅ 2.73

Fig 1.4.10 Three-dimensional Sierpinski gasket composed of full tetrahedra (above left); Menger sponge (above right) (Taken from Mandelbrot, 1982.)

This again illustrates the fact that fractal dimension alone does notcharacterize the object: the fractal dimension of the three-dimensional gasket isequal to two, as is that of the Peano curve

Nonuniform fractals

Another possible type of fractal structure relies on the simultaneous use ofseveral dilation scales Here is an example of such a structure, obtained bydeterministic iteration using factors 1/4 and 1/2 (Fig 1.4.11) This structure isclearly fractal and its dimension D is determined by one iteration, as before,(from Eq 1.4.2):

L/4

L/2L/4

Fig 1.4.11 Construction of a nonuniform deterministic fractal: here with two

scales of contraction.

1.4.2 Random fractals

Up to now only examples of deterministic (also called “exact”) fractalshave been given, but random structures can easily be built In these structuresthe recurrence defining the hierarchy is governed by one or more probabilisticlaws which fix the choice of which generator to apply at each iteration

Homogeneous fractals

A random fractal is homogeneous when the structure’s volume (or mass)

is distributed uniformly at each hierarchical level, that is, the differentgenerators used to construct the fractal keep the same mass ratio from one level

Fig 1.4.12-a Random fractal generator.

the following random fractal may be constructed:

Fig 1.4.12-b Random fractal generated by the previous generator.

The corresponding fractal dimension is

D = d + log β / log 2 = 1

Finding D from a single iteration, we have 2dβ new elements, each of size1/2 at each iteration, thus 2dβ (1/2)D = 1

Heterogeneous fractals

The mass ratio ß may itself vary: a fractal constructed in this way is said to

be heterogeneous (Figs 1.4.13a and 1.4.13b) This type of fractal can be used

as a basis for modeling turbulence (see Sec 2.3)

Starting with a recurrence relation, with a given distribution of ß,

variable mass

(ß=1/4) or (ß=2/4)

Fig 1.4.13-a Generator of a heterogeneous random fractal.

it is possible to build up heterogeneous fractal structures:

…

Fig 1.4.13-b Heterogeneous random fractal generated by the previous fractal.

whose dimension is given by 〈 M(L) 〉 ∝ LD Hence,

D = d + log 〈ß〉 / log 2Random fractals are, with some notable exceptions, almost the only onesfound in nature; their fractal properties (scale invariance, see Sec 1.4.3) bear onthe statistical averages associated with the fractal structure

Example: Fig 1.4.14 below shows a distribution of disks ,the positions of

whose centers follow a Poisson distribution, and whose radii are randomlydistributed according to a probability density, P(R>r) = Q r−α ; the larger thevalue of α, the higher is the frequency of smaller disks, and the further thefractal dimension of the black background is from 2 Such a distribution ofdisks could belong to lunar craters seen from above (projection) or holes in apiece of Emmenthal cheese! We shall be returning to this model later

Fig 1.4.14 Random fractal of discs whose sizes are distributed according to a

fractal structure simply by looking at it: the object appears similar to itself

“from near as from far,” that is, whatever the scale Naturally the eye isinadequate and a more refined analysis is required In previous examples thisinvariance came from the fact that an iterative structure is such that its massobeys a homothetic relation of the form7 (L large)

ordinary surfaces or volumes λ = bd, where d is the dimension of the object.This relation generalizes to all self-similar fractals For the Koch curve, forexample,

M(3L) = 4M(L) = 3DM(L),and in the general case,

M (bL) = bDM (L) (1.4-3)

This is a very direct method of calculating D, which is thus also the similarity

dimension, (see also the remark p 14).

The scale invariance relation M(bL) = bD M(L) is equivalent to the mass/radius

relation M(L) = A0LD To see this, choose b = 1/L in the scaling law, giving M(L) = M(1) LD.

We shall see that internal similarity is present in many exact or randomfractals (see below), which are not generated by iteration

Generally speaking:

Translational invariance → periodic networks

Dilation invariance → self-similar fractals

In practice scale invariance only works for a limited range of distances r:

a << r << Λ

Λ is the macroscopic limit due to the size of the sample, correlation length,effects of gradients, etc and a is the microscopic limit due to the lattice distance,molecular sizes, etc When we come to discuss macroscopic structures inchapter 2 (and microscopic structures in chapter 3), we mean to say that thescales of a and Λ are macroscopic (or microscopic, respectively) Moreover, ifthere are corrections to the scaling law [A(r) not constant], scale invariance willonly be found asymptotically (for very large r)

1.4.4 Ambiguities in practical measurements

In practice, in other words, for physical objects, we find problems inapplying the methods we have just described This is partly because, as wementioned in the last paragraph, there is both a minimum characteristic sizebelow which the fractal description ceases to be valid (for example, aggregatescomposed of small particles) and also an upper size limit for the object underconsideration But it is also because a physical phenomenon, dependent onsome (dominant) parameter, only generates fractal structures on all scales for acritical value of this parameter This value is often difficult to attain, so scalinglaw corrections are generally required (see remark p 14)

The fractal dimension is obtained from the slope of the linear regression ofthe points with coordinates, {log 1/r, log N(r)}, with r going from the minimumcharacteristic size to the size of the object.

The method employed to determine the dimension (discussed by Tricot,1982), the limited extent of a fractal dynamic (object only fractal over a scalespanning less than two orders of magnitude), and the presence of scaling lawcorrections, all contribute to making this method based on log-log regressionsometimes imprecise Such a representation tends to straighten any curves Thepresence of slight curvature is a sign that the asymptotic régime has not beenreached Furthermore, as a point of inflexion may be taken for linear behavior,

it is not always obvious, without reliable theoretical support, what is happening

in this case A good theoretical model or a dynamic that extends over asufficiently wide range of scales is therefore indispensable A discussion aboutreal and apparent power laws may be found in T A Witten, Les Houches,

1985 (Boccara and Daoud, 1985)

M

exp=R + 0 5 R

Mregr 1 33 R1.661

0.751 1.751

1

Fig 1.4.15 Very simple example of imprecision in estimating the exponent (M regr ) due to corrections in the scaling law (M exp ) The value is found to be 1.66 instead 111111111of 1.75, in spite of the seemingly linear nature of the graph.

1.5 Connectivity properties

A fractal is insufficiently characterized by its fractal dimension D, alone.Exponents appear throughout this book with the intension of specifying thebehavior of various physical quantities

Geometrically it is interesting to consider quantities such as the spreading, the ramification, or the lacunarity dimensions, which characterise the properties

of connectivity and of the distribution of matter within a fractal

1.5.1 Spreading dimension, dimension of connectivity

Let us consider a fractal composed of squares lying next to each other (Fig1.5.1) The white squares are permitted sites, the black squares are forbidden

By counting the number S(l) of permitted sites accessible from a given

permitted site in l or less steps, where a step consists of passing to a bordering

permitted site, generally, a mass–(distance in steps) relation of the form,

S(l ) 1 l de (1.5-1)

is obtained l is called the chemical distance or distance of connectivity and dethe spreading dimension This dimension depends solely on the connections

between the elements of the fractal structure (and not on the metric of the space

in which it is embedded): it is an intrinsic connectivity property of the fractal.

The dimension ds is linked to the tortuosity: the higher the tortuosity, the more

the convolutions of the object make us use detours to get from one point toanother situated at a fixed distance “as the crow flies.” We have the inequality

de 1 D The equality, de = D, is attained when the fractal metric corresponds toits natural metric, that is, when the Euclidean dimensions (as the crow flies) areequal or proportional to the distances obtained by staying inside the fractal (onaverage) This is the case for Sierpinski gaskets

Because of the statistical invariance of the structure under any dilation, we

would expect the mean quadratic Euclidean distance R(l )2 between two sites

separated by a “chemical distance” l to be such that

3, from an origin 0.The forbid-

den regions are in black.

The shaded areas

corre-the square root of corre-the mean

square distance between

two points, apart 1l

1l

2 3 4

9

11 10

6 9

7

4 3

4 3

4

4 5

5 6

10 1

l =

R( )l

0

Fig 1.5.1 Representation on a square network of accessible sites, chemical distance

l, and visited sites S(l ) For example, S(3) = 11 = 3 (1)+3 (2)+5 (3).

1.5.2 The ramification R

R is the smallest number of links which must be cut to disconnect a macroscopic part of the object Links should be understood as paths leadingfrom one point of the structure to another

Thus, for the Sierpinski gasket R is finite, while for the carpet R is

infinite Ramification plays an important role in the conduction and

mechanical properties of fractals The condition that R be finite is, moreover,

a necessary condition for exact relations of the renormalization group in real

space This will be proven for the example of a Sierpinski gasket, where itsvibration modes are calculated in Sec 5.1.1 and its conductance in Sec 5.2.1

1.5.3 The lacunarity L

This indicates, in some sense, how far an object is from beingtranslationally invariant, by measuring the presence of sizeable holes in a fractalstructure E

We have seen that it is always possible to write M(R)=A(R) RD, the solecondition on A being that log A/log R!0 The distribution of holes or lacunae

is consequently related to the fluctuations around the law in RD The lacunarity

Fig 1.5.2 Effect of lacunarity on the mass relation as a function of the radius.

M (R) fluctuates around the power law in R D.

examples, respectively This is also true for deterministic fractals obtained by

iterating a generator The lacunarity becomes aperiodic for random fractals(Gefen et al., 1983).

1.6 Multifractal measures

Knowledge of the fractal dimension of a set (as we have seen), isinsufficient to characterize its geometry, and, all the more so, any physicalphenomenon occurring on this set Thus, in a random network of fusible links,the links which melt are those through the current exceeds a certain threshold.Their distribution is supported by a set whose fractal dimension generallydiffers from that of the whole Likewise, in a growth phenomenon, such asdiffusion limited aggregation, (the DLA model is described in Sec 4.2), thegrowth sites do not all have the same weight; some of them grow much morequickly than others Therefore, to understand many physical phenomena,involving fractal supports, the (singular) distribution of measures associatedwith each point of the support must be characterized These measures, scalarquantities, may correspond to concentrations, currents, electrical or chemicalpotentials, probabilities of reaching each point of the support, pressures,dissipations, etc

Intuitive approach to multifractality

Let us take as an example the distribution of diamonds over the surface ofthe earth We shall suppose that the statistics governing this distribution hascertain similarities with the one in Fig 1.4.14 (a little too optimistic !): that is,

we suppose that the distribution of diamonds is fractal, that it is nothomogeneous, and that there exist very few regions where large stones occur:the majority of places on the earth’s surface containing only traces ofdiamonds The information we can draw from knowing the fractal dimension isglobal: if it is close to two, diamonds are spread almost uniformly throughoutthe world, if it is close to zero, there are a few privileged places where all of thediamonds are concentrated In the first situation we soon find something, butthe yield is very low; in the second case we must search for a long time but then

we will be well paid for our efforts We quickly realize that the information

provided by the fractal dimension is inadequate There is a measure attached to

the support of this fractal set, which is the price of diamonds as a function oftheir volume, clearly it is more worthwhile to find large stones than small ones

We must use our knowledge of the distribution of diamonds by bringing in theparameter of their size Let us suppose that we know the distribution perfectly(otherwise we must take a sample) and let us cover the globe with a grid,attaching to each of its square plots of side 6 the monetary value of thediamonds found there To simplify matters the set of plots of land can bedivided into a finite number of batches (i =1, ,N) corresponding to the variousslices of value, from the poorest to the richest [to each plot i we attach in this

way its value µ(6,xi) relative to the total value] The correspondence of each ofthese batches to a given slice µ specifies a distribution of diamonds on theearth's surface; we assume that each of these distributions is fractal in the limitwhen the side 6 of the plots tends to zero The very rare, rich regions will have afractal dimension close to zero (“dust”), whereas the regions with only a trace

of diamonds, albeit uniformly distributed, will have a fractal dimension close totwo

The multifractal character is connected with the heterogeneity of thedistribution (see Sec 1.4.2, and T.A Witten in Les Houches, 1987) For ahomogeneous fractal distribution, the mass in the neighborhood of any point inthe distribution is arranged in the same manner That is to say that inside asphere of radius R centered on the fractal at xi, the mass M(R) fluctuates little

about its mean value over all the xi, #M(R)$ whose scaling law is RD: the

distribution P(M) of the masses M(R) taken at different xi is narrow, that is, itdecreases on either side of the mean value faster than any power In particular,all the moments vary like #M(R)q $ % #M(R)$ q , for all q The fractaldistribution is described by the sole exponent D This is not so for

heterogeneous fractals for which there is a broad distribution, P(M), of the

masses Such is the case for the distribution of diamonds on the earth’ssurface Knowledge of the behavior of the moments #M(R) q $ tells us about theedges of the distribution, namely the very poor and the very rich regions

We are going to make these ideas sharper using some simple distributionswhich will allow us to introduce some mathematical relationships indispensable

in the practical use of the concept of multifractility

The quantities f(3) and 9(q) (which we are now going to define) will allow

us to characterize the distributional heterogeneities of the measures known as

multifractal measures As these ideas are not initially obvious, the reader may,

if he wishes, turn to the various examples given further on

1.6.1 Binomial fractal measure

This simple measure is constructed as follows: a segment of length L, onwhich a uniform measure of density 1/L is distributed, is divided into two parts

of equal length: l0 = l1= L/2 to which the weights p0 and p1 are given (p0 to theleft and p1 to the right) (Fig 1.6.1) This process is iterated ad infinitum Thetotal measure is preserved if care is taken in choosing

p0 + p1 = 1

Each element (segment) of the set is labeled by the successive choices (0left, 1 right) at each iteration At the nt h iteration, each segment is thus indexed

by a sequence [7] ) [70, 71, …, 7k,… 7n] where 7k = 0 or 1, and has length,

dl = 6L = 2– n L Its abscissa on the segment E=[0,1] is described simply bythe number in base two

x = l / L = 0 7071…7n

Fig 1.6.1 First iteration of the binomial measure The measures associated with the

areas of the rectangles are shown inside them.

So, at the third iteration, the successive weights µ(6,xi) are the p[7] where

[7] = [000] (x0=0.000) ! p[000]= p0

[001] (x1=0.001)

[010] (x2=0.010) ! p[001]= p[010]= p[100]= p0 p1[100] (x3=0.100)

[011] (x4=0.011)

[101] (x5=0.101) ! p[011]= p[101]= p[110]= p0p1[110] (x6=0.110)

[111] (x7=0.111) ! p[111]= p1

and so on for each value of n (or of 6 = 2– n) For each n, the distribution isnormalized to one:

1i

To each value of x is associated a 80(x) = 1–81(x) These weights occurwith a frequency

N(1,x) = n

k =

n!

(n20)!(n21)! Fig 1.6.2 shows the hierarchy of iterations for n = 4, and the correspondingdistribution of weights (more precisely their logarithm) The logarithm of theweight on an interval 6, divided by the logarithm of that interval, is called the

Holder exponent and is denoted by 3 ,

1 =log µ(2)log 2 = – 30log2p0 – 31log2p1

It measures the singular behavior of the measure in the neighborhood of a

point x [via 80(x) and 81(x)], thus

The values of 3 are bounded according to the following inequality:

0 < 3min= –log2 p0 2 3 2 3max= –log2 p1 < 1

Fig 1.6.2 (Left) the binary sequences leading to the [7] for n=4 The graph on the right shows the distribution of the measures µ whose logarithm is also proportional (factor –n) to 3 Here we have chosen log p 1 = 2 log p 0 We can see the symmetry

of the binomial distribution: contrary to the example of the diamonds, the regions of 111111111111low weight are as rare as those with high weight.

This behavior allows us to partition8 the set E into subsets having the same 3,

1 = 32

12

Let us now take the subsets E3 into closer consideration We enumeratedthem above while calculating N(6) The analog of the Hausdorff dimension forthe support of intervals with the same 3 (function of 3 and therefore of x) isthen written

8 In our intuitive example about the distribution of diamonds the partition into “batches” is made according to their relative “value;” we see here that these batches may also be designated by their Holder exponents which indicates how the value of a piece of land varies (locally) as a function of its size.

1.6.2 Multinomial fractal measure

The results just shown for a binomial measure easily generalize to

multinomial measures By considering b weights p4 (0 2 4 2 b–1), we can gothrough an analogous procedure to that of the binomial measure Each segment

of size b– n at iteration n is indexed by a sequence [7] or an abscissa x, written

in base b (instead of the 0’s and 1’s of the previous binomial example) The adic intervals are then characterized by the frequencies 84 of their “digits” inbase b So, the expressions for 3 and 5 generalize to

with the constraints 1 and

In the case of a binomial measure (b = 2), 5 is a single valued function of

3, since two parameters (80 and 81), whose sum is normalized to one, are used.For b > 2, this relationship is no longer single valued (there are b–2supplementary parameters) and the pairs (3,5) cover a certain domain Thisdomain is roughly indicated by a network of curves in Fig 1.6.3 (for b = 3).The set of [7] (or x expressed in base b) corresponding to the same 3 isdominated by the term of highest dimension, f = max 5 [i.e., by the subsetN(6,3) whose exponent 5 is the greatest]:

N(2, 1)dominant 3 2– f(1) (1.6-4)This term therefore maximizes

This is solved in the classical way by introducing Lagrange multipliers q and(r–1), leading to the relation

•Figure a shows the distribution of the

4 for b2=23 (25 values for 6 and 6 , 6 being held fixed).

• Figure bshows the curve of maxima in f

It vanishes for 2 = – log p 1

2 = – log p 1 4.19 The maximum of f

is attained when 2 = – 1 p log p 1 2.59.

• Figure c : For a set of values p , p ,

and p close to 1/3 , the distribution µ (5)

is nearly uniform, and f (2) is sharply peaked about the value 21= 1.

Fig 1.6.3 Distribution of the Hausdorff dimensions of subsets E3

It is then useful to introduce the quantity